Project: Generating Fractal Geometry

Click to see gallery of full-sized images…

Introduction

During the development of my PlotEquation plugin, I was introduced to self-similar mathematical objects known as fractals. Unlike the explicit expressions I was used to, which iterate through space to see where the function is satisfied, these objects are generated by recursively iterating a set of user-defined rules. Given the limitless variety in fractal objects and that their beauty cannot be emulated with explicit expressions, I expanded the scope of my plugin to include fractal objects as well. A more expansive gallery with more kinds of fractal geometry can be found here.

Gallery

• Naming Key: Fractal Name (Iteration #): Specifications

• Note that “Filled” refers to the filling of the central polygon for Sierpinski objects.

Koch Curve (6); Angle: 60°, Type: I

Koch Curve (6); Angle: 85°, Type: I

Koch Curve (5); Angle: 85°, Type: II

Koch Curve (5); Angle: 85°, Type: III

Koch Curve (5); Angle: 45°, Type: I

Koch Curve (5); Angle: 45°, Type: II

Koch Curve (5); Angle: 45°, Type: III

Koch Curve (5); Angle: 15°, Type: I

Koch Curve (5); Angle: 15°, Type: II

Koch Curve (5); Angle: 15°, Type: III

Koch Snowflake (5); Sides: 3, Angle: 60°, Type: I

Koch Snowflake (5); Sides: 3, Angle: 60°, Type: II

Koch Snowflake (5); Sides: 3, Angle: 60°, Type: III

Koch Snowflake (5); Sides: 3, Angle: 45°, Type: II

Koch Snowflake (5); Sides: 4, Angle: 45°, Type: III

Koch Snowflake (5); Sides: 5, Angle: 30°, Type: II

Koch Snowflake (5); Sides: 5, Angle: 80°, Type: III

3D Koch Snowflake (5): Cube Method

3D Koch Snowflake (5): Delta Method

Sierpinski Triangle (5)

Inverse Sierpinski Triangle (5)

Sierpinski Triangle (7)

Inverse Sierpinski Triangle (7)

Sierpinski Triangle (10)

Inverse Sierpinski Triangle (10)

Sierpinski Triangle (6); Skewed

Inverse Sierpinski Triangle (6); Skewed

Sierpinski Triangle (5); Filled

Sierpinski Square (5)

Sierpinski Pentagon (5)

Sierpinski Pentagon (5); Filled

Sierpinski Hexagon (5)

Sierpinski Hexagon (5); Filled

Sierpinski Heptagon (5)

Sierpinski Heptagon (5); Filled

Sierpinski Octagon (5)

Sierpinski Octagon (5); Filled

Sierpinski Dodcagon (5)

Sierpinski Dodecagon (5); Filled

Sierpinski Tetrahedron (5)

Inverse Sierpinski Tetrahedron (5)

Sierpinski Tetrahedron (10)

Sierpinski Tetrahedron (5); Skewed

Sierpinski Pyramid (5)

Sierpinski Pyramid (5); Skewed

Sierpinski Octahedron (5)

Inverse Sierpinski Tetrahedron (10)

Inverse Sierpinski Tetrahedron (5); Skewed

Sierpinski Pyramid (7)

Sierpinski Pyramid (7); Skewed

Sierpinski Octahedron (7)

Sierpinski Octahedron (5); Skewed